As we noted in Chapter 1, the Chicago Board of Trade was established to provide farmers, dealers, and food processors a way of hedging against price risk by entering forward contracts to buy or sell a commodity at a future date at a price specified today. While futures contracts enable businesses, farmers, and other economic entities to hedge the costs or revenues from unfavorable price movements, they also eliminate the benefits realized from favorable price movements. One of the differences of using options instead of futures contracts as a hedging tool is that the hedger, for the price of the option, can obtain protection against adverse price movements while still realizing benefits if the underlying asset moves in a favorable direction. Some of the important uses of options in hedging are from purchasing stock, index, futures options, and OTC put options as a way of attaining downside protection on the future sale of a portfolio, currency, bond, or commodity and by purchasing call options as a strategy for capping the costs of future purchases. In this chapter, we examine hedging equity, commodity, currency, and fixed‐income positions with spot and futures options.

Hedging Stock Portfolio Positions

Creating a Floor for a Stock Portfolio Using Index Options

An equity portfolio insurance strategy is a hedging position in which an equity portfolio manager protects the future value of her fund by buying spot or futures index put options. The index put options, in turn, provide downside protection against a stock market decline, while allowing the fund to grow if the market increases. As an example, consider an equity fund manager who on May 27, 2016, planned to sell a portion of a stock portfolio in mid‐August to meet an anticipated liquidity need. Suppose the portfolio the manager planned to sell was well‐diversified and highly correlated with the S&P 500, had a beta (β) of 1.25, and currently was worth V₀ = $100 million. On May 27, the spot S&P 500 was at 2,100. Suppose the manager on that date expected a bullish market to prevail in the future with the S&P 500 rising. As a result, the manager would have been hoping to benefit from selling her portfolio in August at a higher value. At the same time, though, suppose the manager was also concerned that the market could be lower in mid‐August and did not want to risk selling the portfolio in a market with the index lower than 2,100. On May 27, an August S&P 500 put option with an exercise price of 2,100 and multiplier of 100 was trading at 54.40. As a strategy to lock in a minimum value from the portfolio sale if the market decreased, while obtaining a higher portfolio value if the market increased, suppose the manager set up a portfolio insurance strategy by buying August S&P 500 2,100 puts. To form the portfolio insurance position, the manager would have needed to buy 595.2381 August S&P 500 puts (assume perfect divisibility) at a cost of $3,238,095:

Exhibit 8.1 shows for spot index values ranging from 1,860 to 2,460, the manager’s revenue that would have resulted from selling the portfolio at the option’s August expiration date and closing her puts by selling them at their intrinsic values. Note, for each index value shown in Column 1, there is a corresponding portfolio value (shown in Column 4) that reflects the proportional change in the market given the portfolio beta of 1.25. For example, if the spot index were at 1,860 at expiration, then the market as measured by the proportional change in the index would have decreased by 11.43% from its May 27 level of 2,100 (−0.1143 = (1,860 − 2,100)/2,100). Since the well‐diversified portfolio has a beta of 1.25, it would have decreased by 14.29% (β (%Δ S&P 500) = 1.25(−0.1143) = −0.1429), and the portfolio would, in turn, have been worth only $85,714,280—85.714% of its May 26 value of $100 million. Thus, if the market were at 1,860, the corresponding portfolio value would be $85,714,286 (= (1 + β (%Δ S&P 500)V₀ = (1 + 1.25(−0.1143)($100,000,000))). On the other hand, if the spot S&P 500 index were at 2,460 at the August expiration, then the market would have increased by 17.14% (= (2,460 − 2,100)/2,100) and the portfolio would have increased by 21.43% (=1.25(0.1714)) to equal $121,428,571 (=1.2143 ($100,000,000)). Thus, when the market is at 2,460, the portfolio’s corresponding value is $121,428,571. Given the corresponding portfolio values, Column 5 in Exhibit 8.1 shows the intrinsic values of the S&P 500 put corresponding to the spot index values, and Column 6 shows the corresponding cash flows that would be received by the portfolio manager from selling the 595.2381 expiring August index puts at their intrinsic values. As shown in the exhibit, if the spot S&P 500 is less than 2,100 at expiration, the manager would have realized a positive cash flow from selling her index puts, with the put revenue increasing proportional to the proportional decreases in the portfolio values, providing, in turn, the requisite protection in value. On the other hand, if the S&P 500 spot index is equal to or greater than 2,100, the manager’s put options would be worthless, but her revenue from selling the portfolio would be greater, the greater the index. Thus, if the market were 2,100 or less at expiration, the value of the hedged portfolio (stock portfolio value plus put values) would be $100 million; if the market were above 2,100, the value of the hedged portfolio would increase as the market rises. Thus, for the $3,238,095 cost of the put options, the fund manager would have attained on May 27 a $100 million floor for the value of the portfolio in mid‐August, while benefiting with greater portfolio values if the market increased.

Portfolio Hedged with S&P 500 Index Puts; Portfolio: Initial Value = $100,000,000, β = 1.25; S&P 500 Put: X = 2100, Multiplier = 100, Premium = 54.40; Hedge: 595.2381 Puts; Cost = (595.2381)(54.40)($100) = $3,238,095 Short S&P 500 Call: X = 2100, Multiplier = 100, Premium = 24.10; Hedge: 595.2381; Revenue = (595.2381)(24.10)($100) = $1,434,524 Range Forward: Long 595.2381 Put Contracts and Short 595.2381 Call Contracts; Net cost = $3,328,095 − $1,434,524 = $1,893,571
1	2	3	4	5	6	7	8	9	10
S&P 500 at T, ST	Proportional Change in S&P 500: g = (ST − 2100)/2100	Proportional Change in Portfolio: βg = 1.25g	Portfolio Value: VT = (1+βg)$100m	Put Value PT = IV = Max[2100 − ST,0]	Value of Puts: CF = (595.2381)($100) (IV)	Hedged Portfolio Value	Short Call Value: IV = Max[ST − 2310,0]	Value of Short Call: 595.2381(100)IV	Ranged Forward: Hedged Portfolio Value = (4) + (6) − (9)
1860	−0.1143	−0.1429	$85,714,286	240	$14,285,714	$100,000,000	$0	$0	$100,000,000
1890	−0.1000	−0.1250	$87,500,000	210	$12,500,000	$100,000,000	$0	$0	$100,000,000
1920	−0.0857	−0.1071	$89,285,714	180	$10,714,286	$100,000,000	$0	$0	$100,000,000
1950	−0.0714	−0.0893	$91,071,429	150	$8,928,571	$100,000,000	$0	$0	$100,000,000
1980	−0.0571	−0.0714	$92,857,143	120	$7,142,857	$100,000,000	$0	$0	$100,000,000
2010	−0.0429	−0.0536	$94,642,857	90	$5,357,143	$100,000,000	$0	$0	$100,000,000
2040	−0.0286	−0.0357	$96,428,571	60	$3,571,429	$100,000,000	$0	$0	$100,000,000
2070	−0.0143	−0.0179	$98,214,286	30	$1,785,714	$100,000,000	$0	$0	$100,000,000
2100	0.0000	0.0000	$100,000,000	0	$0	$100,000,000	$0	$0	$100,000,000
2130	0.0143	0.0179	$101,785,714	0	$0	$101,785,714	$0	$0	$101,785,714
2160	0.0286	0.0357	$103,571,429	0	$0	$103,571,429	$0	$0	$103,571,429
2190	0.0429	0.0536	$105,357,143	0	$0	$105,357,143	$0	$0	$105,357,143
2220	0.0571	0.0714	$107,142,857	0	$0	$107,142,857	$0	$0	$107,142,857
2250	0.0714	0.0893	$108,928,571	0	$0	$108,928,571	$0	$0	$108,928,571
2280	0.0857	0.1071	$110,714,286	0	$0	$110,714,286	$0	$0	$110,714,286
2310	0.1000	0.1250	$112,500,000	0	$0	$112,500,000	$0	$0	$112,500,000
2340	0.1143	0.1429	$114,285,714	0	$0	$114,285,714	$30	$1,785,714	$112,500,000
2370	0.1286	0.1607	$116,071,429	0	$0	$116,071,429	$60	$3,571,429	$112,500,000
2400	0.1429	0.1786	$117,857,143	0	$0	$117,857,143	$90	$5,357,143	$112,500,000
2430	0.1571	0.1964	$119,642,857	0	$0	$119,642,857	$120	$7,142,857	$112,500,000
2460	0.1714	0.2143	$121,428,571	0	$0	$121,428,571	$150	$8,928,572	$112,500,000

Creating a Cap for a Stock Portfolio Purchase Using Index Options

In addition to protecting the value of a portfolio, spot and futures index options also can be used to hedge the costs of future stock portfolio purchases. For example, suppose on May 27 the above portfolio manager was anticipating a cash inflow of $100 million in August, which she planned to invest in a well‐diversified portfolio with a β of 1.25 that was currently worth $100 million when the current spot S&P 500 index was at 2,100. On May 27, an August S&P 500 call option with an exercise price of 2,100 and multiplier of 100 was trading at 45.70. As a strategy to cap the portfolio purchase in case the market increases, the manager could have purchased 595.2381 August S&P 500 2100 calls (assume perfect divisibility) at a cost of $2,720,238:

As shown in Exhibit 8.2, if the spot index were 2,100 or higher at the August expiration, the corresponding cost of the portfolio would be higher; the higher portfolio costs, though, would have been offset by profits from the index calls. For example, if the market were at 2,370 in September, then the well‐diversified portfolio with a β of 1.25 would cost $121,428,571; the additional $16,071,429 cost of the portfolio would be offset, though, by the $16,071,429 cash flow obtained from the selling 595.2381 August 2,100 index calls at their intrinsic value of 270. Thus, as shown in the exhibit, for index values of 2,100 or greater, the hedged costs of the portfolio would be $100 million. On the other hand, if the index is less than 2,100, the manager would have been able to buy the well‐diversified portfolio at a lower cost, with the losses on the index calls limited to just the premium. Thus, for the $2,720,238 costs of the index call option, the manager on May 27 could have capped the maximum cost of the portfolio in mid‐August at $100 million, while still benefiting with lower costs if the market declined.

Portfolio Purchase with Cost Hedged with S&P 500 Index Calls; Portfolio: Initial Value at Time of Hedge = $100,000,000, β = 1.25 S&P 500 Call: X = 2,100, Multiplier = 100, Premium = 45.70; Hedge: 595.2381 Calls; Cost = (595.2381)(45.70)($100) = $2,720,238 Short S&P 500 Put: X = 1,910, Multiplier = 100, Premium = 24.10; Hedge: 595.2381; Revenue = (595.2381)(24.10)($100) = $1,434,524 Range Forward: Long 595.2381 Call Contracts and Short 595.2381 Put Contracts; Net Cost = $2,720,238 − $1,434,524 = $1,285,714
1	2	3	4	5	6	7	8	9	10
S&P 500 at T, ST	Proportional Change in S&P 500: g = (ST − 2100)/2100	Proportional Change in Portfolio: βg = 1.25g	Portfolio Cost: VT = (1+βg)$100m	Call Value CT = IV = Max[ST − 2100,0]	Value of Calls: CF = (595.2381)($100) (IV)	Hedged Portfolio Cost: Col (4) − Col (6)	Short Put Value: = IV = Max[1910 − ST, 0]	Value of Short Put: 595.2381(100) IV	Range Forward: Hedged Portfolio Cost: (4) − (6) − (9)
1770	−0.1571	−0.1964	$80,357,143	0	$0	$80,357,143	−$140	−$8,333,333	$88,690,476
1800	−0.1429	−0.1786	$82,142,857	0	$0	$82,142,857	−$110	−$6,547,619	$88,690,476
1830	−0.1286	−0.1607	$83,928,571	0	$0	$83,928,571	−$80	−$4,761,905	$88,690,476
1860	−0.1143	−0.1429	$85,714,286	0	$0	$85,714,286	−$50	−$2,976,191	$88,690,476
1890	−0.1000	−0.1250	$87,500,000	0	$0	$87,500,000	−$20	−$1,190,476	$88,690,476
1910	−0.0905	−0.1131	$88,690,476	0	$0	$88,690,476	$0	$0	$88,690,476
1920	−0.0857	−0.1071	$89,285,714	0	$0	$89,285,714	$0	$0	$89,285,714
1950	−0.0714	−0.0893	$91,071,429	0	$0	$91,071,429	$0	$0	$91,071,429
1980	−0.0571	−0.0714	$92,857,143	0	$0	$92,857,143	$0	$0	$92,857,143
2010	−0.0429	−0.0536	$94,642,857	0	$0	$94,642,857	$0	$0	$94,642,857
2040	−0.0286	−0.0357	$96,428,571	0	$0	$96,428,571	$0	$0	$96,428,571
2070	−0.0143	−0.0179	$98,214,286	0	$0	$98,214,286	$0	$0	$98,214,286
2100	0.0000	0.0000	$100,000,000	0	$0	$100,000,000	$0	$0	$100,000,000
2130	0.0143	0.0179	$101,785,714	30	$1,785,714	$100,000,000	$0	$0	$100,000,000
2160	0.0286	0.0357	$103,571,429	60	$3,571,429	$100,000,000	$0	$0	$100,000,000
2190	0.0429	0.0536	$105,357,143	90	$5,357,143	$100,000,000	$0	$0	$100,000,000
2220	0.0571	0.0714	$107,142,857	120	$7,142,857	$100,000,000	$0	$0	$100,000,000
2250	0.0714	0.0893	$108,928,571	150	$8,928,571	$100,000,000	$0	$0	$100,000,000
2280	0.0857	0.1071	$110,714,286	180	$10,714,286	$100,000,000	$0	$0	$100,000,000
2310	0.1000	0.1250	$112,500,000	210	$12,500,000	$100,000,000	$0	$0	$100,000,000
2340	0.1143	0.1429	$114,285,714	240	$14,285,714	$100,000,000	$0	$0	$100,000,000
2370	0.1286	0.1607	$116,071,429	270	$16,071,429	$100,000,000	$0	$0	$100,000,000

Portfolio Exposure—Market Timing and Beta Convexity

Instead of hedging a portfolio’s value against market risk, suppose a manager wanted to change her portfolio’s exposure to the market. As we discussed in Chapter 3, an equity portfolio manager who is very confident of a bull (bear) market can increase (decrease) her portfolio’s exposure to the market by increasing (decreasing) the portfolio’s beta, β₀, to a new target beta, β_TR, by going long (short) in equity index futures contracts. The manager also could increase (decrease) her portfolio’s exposure by buying index calls (puts). The number of option contracts needed to move the portfolio beta from β₀ to β_TR can be determined using the price‐sensitivity model in which:

where

• if β_TR > β₀, Long equity index call options
• if β_TR < β₀, Long equity index put options

Changing a portfolio’s market exposure with options instead of futures provides an asymmetrical gain and loss relation, referred to as a convex beta. For example, when calls (puts) are purchased to increase (decrease) the target beta, the option‐adjusted portfolio has a β_TR for market increases (decreases) and β₀ for market decreases (increases). The cost of obtaining this asymmetrical or convex beta relation is the cost of the options.

For example, suppose on May 27, 2016, the equity fund manager in our earlier example was very bullish and wanted to increase her beta from 1.25 to 2.00. On May 27, the portfolio had a value of V₀ = $100 million, and there was an August S&P 500 call with an exercise price of 2,100 trading at 45.70. To increase the portfolio’s beta to 2.00, the manager would have needed to buy 357.142857 August S&P 500 calls for $1,632,143:

As shown in Exhibit 8.5, if the market increases from 2,100, the manager earns higher proportional gains from the call‐enhanced portfolio than from the unadjusted portfolio. As shown in the exhibit, the proportional increases in the call‐enhanced portfolio for proportional increases in the market reflect a beta of 2.00. On the other hand, if the market decreases from 2,100, then proportional declines in the portfolio for declines in the market reflect a beta of 1.25.

Portfolio Enhanced with S&P 500 Index Calls; Portfolio: Initial Value = $100,000,000, β = 1.25; Target: βTR = 2.00 S&P 500 Call: X = 2,100, Multiplier = 100, Premium = 45.70; Hedge: 357.142857 Calls; Cost = (357.142857)(45.70)($100) = $1,632,143
1	2	3	4	5	6	7	8	9
S&P 500 at T, ST	Proportional Change in S&P 500: g = (ST − 2100)/2100	Proportional Change in Portfolio: βg = 1.25g	Portfolio Value: VT = (1+βg)$100m	Call Value CT = IV = Max[ST − 2,100,0]	Value of Calls: CF = (357.142857)($100) (IV)	Enhance Hedged Value: Col (4) + Col (6)	Proportional Change: [Col (7)/$100m)] − 1	Beta: Col (8)/Col (2)
1,910	−0.0905	−0.1131	$88,690,476	0	$0	$88,690,476	−0.1131	1.2500
1,940	−0.0762	−0.0952	$90,476,190	0	$0	$90,476,190	−0.0952	1.2500
1,970	−0.0619	−0.0774	$92,261,905	0	$0	$92,261,905	−0.0774	1.2500
2,000	−0.0476	−0.0595	$94,047,619	0	$0	$94,047,619	−0.0595	1.2500
2,030	−0.0333	−0.0417	$95,833,333	0	$0	$95,833,333	−0.0417	1.2500
2,060	−0.0190	−0.0238	$97,619,048	0	$0	$97,619,048	−0.0238	1.2500
2,090	−0.0048	−0.0060	$99,404,762	0	$0	$99,404,762	−0.0060	1.2500
2,100	0.0000	0.0000	$100,000,000	0	$0	$100,000,000	0.0000	1.2500
2,120	0.0095	0.0119	$101,190,476	20	$714,286	$101,904,762	0.0190	2.0000
2,150	0.0238	0.0298	$102,976,190	50	$1,785,714	$104,761,905	0.0476	2.0000
2,180	0.0381	0.0476	$104,761,905	80	$2,857,143	$107,619,048	0.0762	2.0000
2,210	0.0524	0.0655	$106,547,619	110	$3,928,571	$110,476,190	0.1048	2.0000
2,240	0.0667	0.0833	$108,333,333	140	$5,000,000	$113,333,333	0.1333	2.0000
2,270	0.0810	0.1012	$110,119,048	170	$6,071,429	$116,190,476	0.1619	2.0000
2,300	0.0952	0.1190	$111,904,762	200	$7,142,857	$119,047,619	0.1905	2.0000
2,330	0.1095	0.1369	$113,690,476	230	$8,214,286	$121,904,762	0.2190	2.0000
2,360	0.1238	0.1548	$115,476,190	260	$9,285,714	$124,761,905	0.2476	2.0000
2,390	0.1381	0.1726	$117,261,905	290	$10,357,143	$127,619,048	0.2762	2.0000
2,420	0.1524	0.1905	$119,047,619	320	$11,428,571	$130,476,190	0.3048	2.0000
2,450	0.1667	0.2083	$120,833,333	350	$12,500,000	$133,333,333	0.3333	2.0000

In contrast, suppose on May 27, the equity fund manager was very bearish and wanted to decrease her beta from 1.25 to 0.75. On May 27, the August S&P 500 put with an exercise price of 2,100 was trading at 54.40. To decrease the portfolio’s beta to 0.75, the manager would have needed to buy 238.095238 August S&P 500 puts for $1,295,238:

As shown in Exhibit 8.6, if the market decreases from 2,100, the manager earns smaller proportional losses from the put‐hedged portfolio than from the unadjusted portfolio. As shown in the exhibit, the proportional decreases in the put‐hedged portfolio for proportional decreases in the market reflect a beta of 0.75. On the other hand, if the market increases from 2,100, then proportional increases in the portfolio for increases in the market reflect a beta of 1.25.

Portfolio Hedged with S&P 500 Index Puts; Portfolio: Initial Value = $100,000,000, β = 1.25; Target: βTR = 0.75 S&P 500 Put: X = 2,100, Multiplier = 100, Premium = 54.40; Hedge: 238.095238 Puts; Cost = (238.095238)(54.40)($100) = $1,295,238
1	2	3	4	5	6	7	8	9
S&P 500 at T, ST	Proportional Change in S&P 500: g = (ST − 2100)/2100	Proportional Change in Portfolio: βg = 1.25g	Portfolio Value: VT = (1+βg)$100m	Put Value PT = IV = Max[2,100 − ST,0]	Value of put: CF = (238.095238)($100) (IV)	Put Hedged Value: Col (4) + Col (6)	Proportional Change: [Col (7)/$100m)] − 1	Beta: Col (8)/Col (2)
1,910	−0.0905	−0.1131	$88,690,476	190	$4,523,810	$93,214,286	−0.0679	0.7500
1,940	−0.0762	−0.0952	$90,476,190	160	$3,809,524	$94,285,714	−0.0571	0.7500
1,970	−0.0619	−0.0774	$92,261,905	130	$3,095,238	$95,357,143	−0.0464	0.7500
2,000	−0.0476	−0.0595	$94,047,619	100	$2,380,952	$96,428,571	−0.0357	0.7500
2,030	−0.0333	−0.0417	$95,833,333	70	$1,666,667	$97,500,000	−0.0250	0.7500
2,060	−0.0190	−0.0238	$97,619,048	40	$952,381	$98,571,429	−0.0143	0.7500
2,090	−0.0048	−0.0060	$99,404,762	10	$238,095	$99,642,857	−0.0036	0.7500
2,100	0.0000	0.0000	$100,000,000	0	$0	$100,000,000	0.0000	0.7500
2,120	0.0095	0.0119	$101,190,476	0	$0	$101,190,476	0.0119	1.2500
2,150	0.0238	0.0298	$102,976,190	0	$0	$102,976,190	0.0298	1.2500
2,180	0.0381	0.0476	$104,761,905	0	$0	$104,761,905	0.0476	1.2500
2,210	0.0524	0.0655	$106,547,619	0	$0	$106,547,619	0.0655	1.2500
2,240	0.0667	0.0833	$108,333,333	0	$0	$108,333,333	0.0833	1.2500
2,270	0.0810	0.1012	$110,119,048	0	$0	$110,119,048	0.1012	1.2500
2,300	0.0952	0.1190	$111,904,762	0	$0	$111,904,762	0.1190	1.2500
2,330	0.1095	0.1369	$113,690,476	0	$0	$113,690,476	0.1369	1.2500
2,360	0.1238	0.1548	$115,476,190	0	$0	$115,476,190	0.1548	1.2500
2,390	0.1381	0.1726	$117,261,905	0	$0	$117,261,905	0.1726	1.2500
2,420	0.1524	0.1905	$119,047,619	0	$0	$119,047,619	0.1905	1.2500
2,450	0.1667	0.2083	$120,833,333	0	$0	$120,833,333	0.2083	1.2500

Hedging Currency and Commodity Positions

Hedging Fixed‐Income Positions with Options

As examined in Chapter 4, a fixed‐income manager planning to invest a future inflow of cash in high‐quality, intermediate‐term bonds could hedge the investment against possible higher bond prices and lower rates by going long in T‐note futures contracts. If intermediate‐term rates were to decrease, the higher costs of purchasing the bonds would then be offset by profits from his futures positions. On the other hand, if rates increased, the manager would benefit from lower bond prices, but he would also have to cover losses on his futures position. Thus, hedging future fixed‐income investments with futures locks in a future price and return and therefore eliminates not only the costs of unfavorable price movements but also the benefits from favorable movements. However, by hedging with either exchange‐traded futures call options on a T‐note, T‐bond, Eurodollar deposit, or with an OTC spot call option on a debt security, a hedger can obtain protection against adverse price increases while still realizing lower costs if security prices decrease.

For cases in which bond or money market managers are planning to sell some of their securities in the future or who want to hedge their security values, hedging can be done by going short in a T‐note, T‐bond, or Eurodollar futures contracts. If rates were higher at the time of the sale, the resulting lower bond prices and therefore revenue from the bond sale would be offset by profits from the futures positions (just the opposite would occur if rates were lower). The hedge also can be set up by purchasing an exchange‐traded futures put options on Treasuries and Eurodollar contracts or an OTC spot put option on a debt security. This hedge would provide downside protection if bond prices decrease while earning values if security prices increase.

Short hedging positions with futures and put options can be used not only by holders of fixed‐income securities planning to sell their instruments before maturity, but also by bond issuers, borrowers, and debt security underwriters. A company planning to issue bonds or borrow funds from a financial institution at some future date, for example, could hedge the debt position against possible interest rate increases by going short in debt futures contracts or cap the loan rate by buying an OTC put or exchange‐traded futures put. Similarly, as we examined in Chapter 4, a bank that finances its short‐term loan portfolio of one‐year loans by selling 90‐day CDs could manage the resulting maturity gap by also taking short positions in Eurodollar futures or futures options. Finally, an underwriter or a dealer who is holding a debt security for a short‐period of time could hedge the position against interest rate increases by going short in an appropriate futures contract or by purchasing a futures put option.

Note that many debt and fixed‐income positions involve securities and interest rate positions in which a futures contract on the underlying security does not exist. In such cases, an effective cross hedge needs to be determined to minimize the price risk in the underlying spot position. As noted in Chapter 4, one commonly used model for bond and debt positions is the price‐sensitivity model developed by Kolb and Chiang and Toevs and Jacobs. For option hedging, the number of options (call for long hedging positions and puts for short hedging positions) using the price‐sensitivity model is:

where

Hedging a T‐Note Purchase with T‐Note Futures Calls

In Chapter 4, we presented the case of a fixed‐income manager who on 12/30/15 planned to buy 10 five‐year T‐notes in June from an anticipated $1 million cash inflow resulting from maturing bonds in her portfolio. Concerned about rates decreasing and bond price rising over the next six months, the manager hedged the purchase by going long in 10 CME June five‐year T‐note futures contracts at a futures price of 117.6875. The most likely‐to‐deliver bond on the contract was a T‐note with a 1³/₈% coupon, maturity of 8/31/20, conversion factor (CFA) of 0.8371, and accrued interest on the delivery date of $0.457 (see Exhibit 4.16). With this long futures position, the manager was able to lock in a T‐note cost of $983,557 by purchasing the delivered bonds on the contract and paying the accrued interest:

T‐Note Price per $100 Face Value:

Cost of 10 T‐notes with $100,000 face value plus accrued interest:

Alternatively, the manager could have a realized the $983,557 cost by buying her T‐notes on the spot at the futures expiration and closing her expiring futures position.

Suppose on 12/30/15, the manager wanted to still hedge against the possibility of rates decreasing, but believed that rates would increase, causing five‐year T‐note prices to fall. In this case, the manager, for the cost of 10 June five‐year T‐notes call futures contracts, could cap the June cost of purchasing 10 T‐notes in June while benefiting with lower bond cost if bond prices decreased as she expected. On 12/30/15, there was a CME call on the June five‐year T‐note with an exercise price of 118 selling for $750 per contract. Suppose that the manager purchased the call options in order to set a cap on the cost of buying the five‐year notes in June. Exhibit 8.11 shows:

1. The hedged cost of buying the 10 T‐notes with a 1³/₈% coupon and maturity of 8/31/20 at different spot prices between S_T = 92 and 105.
2. The corresponding June futures prices, where the futures prices are equal to the prices of the cheapest‐to‐deliver 1³/₈% note divided by underlying futures contract’s conversion factor of 0.8371 (f_T = (S_T + AI)/CFA; AI = 0.303 on the option’s expiration of 5/20/16).
3. The cash flow from selling ten 118 call options on the June futures contract at their intrinsic value.
4. The call hedged costs of buying the 1³/₈% T‐notes on the option expiration and selling the calls at their IV.

Cheapest‐to‐Deliver T‐Note: 1 3/8 8/31/20: On 5/20/16 (Futures Expiration), Accrued Interest (AI) = $0.303 per $100 face; Conversion Factor = CFA = 0.8317 Futures Price at T: (Cheapest‐to‐Deliver T‐Note Price + AI)/Conversion Factor = (ST + AI)/CFA Call Options on June 5‐Year T‐Note Futures with X = 118, Expiration = 5/20/15, Premium on 12/30/15 = 0 − 48; [(48/64)/100]($100,000) = $750
1	2	3	4	5	6	7	8
YTM	Price of T‐Note: 1 3/8 8/31/20, ST	Cost of 10 T‐Notes with $100,000 Face plus Accrued Interest: 10 [(ST + $0.303)/100] $100,000	Futures Prices at T, fT: (ST + AI)/(CFA) = (ST + AI)/(0.8317)	Call Value CT = IV = Max[fT − 118,0]	Cash Flow from Selling 10 June Futures Calls at IV: (10) (IV/100) ($100,000)	Futures Call Hedge: Cost of Buying T‐Notes minus Cash Flow from Futures Call: (3) − (6)	Hedged Cost Savings: (3) − (9)
3.39891%	92	$923,030	110.98	$0.00	$0	$923,030	$0
3.13518%	93	$933,030	112.18	$0.00	$0	$933,030	$0
2.87466%	94	$943,030	113.39	$0.00	$0	$943,030	$0
2.89617%	95	$953,030	114.59	$0.00	$0	$953,030	$0
2.63588%	96	$963,030	115.79	$0.00	$0	$963,030	$0
2.37866%	97	$973,030	116.99	$0.00	$0	$973,030	$0
2.12446%	98	$983,030	118.20	$0.20	$1,953	$981,077	$1,953
1.84476%	98.1406	$984,436	118.36	$0.36	$3,643	$980,793	$3,643
1.87319%	99	$993,030	119.40	$1.40	$13,976	$979,054	$13,976
1.62480%	100	$1,003,030	120.60	$2.60	$26,000	$977,030	$26,000
1.37924%	101	$1,013,030	121.80	$3.80	$38,023	$975,007	$38,023
1.36426%	102	$1,023,030	123.00	$5.00	$50,047	$972,983	$50,047
0.89632%	103	$1,033,030	124.21	$6.21	$62,070	$970,960	$62,070
0.65885%	104	$1,043,030	125.41	$7.41	$74,094	$968,936	$74,094
0.42348%	105	$1,053,030	126.61	$8.61	$86,118	$966,912	$86,118

As shown in the exhibit, the hedge caps the upper cost between $966,912 and $981,077 if bond prices are above 98.1406 (futures price of 118) and yields are less than 1.84476%, while allowing the manager to purchase bonds at lower prices if the bond price is at 98.1406 or less and the yield is at 1.84476% or greater.

Hedging a Bond Portfolio with T‐Bond Futures Puts

In Chapter 4, we examined the case of the managers of the Xavier Bond Fund hedging the June value of their portfolio on 12/30/15 by going short in the June 2016 five‐year T‐note futures contract. The fund’s portfolio consisted of 39 investment‐grade corporates, treasuries, and federal agency bonds. On 3/15/16, the value of the bond portfolio was $11,637,935 with a duration of 5.31 and yield to maturity of 2.78% (see Exhibit 4.21 for portfolio description on 12/30/15). Suppose on 3/15/15, the managers believed that rates would decrease and bond prices would rise, but still wanted to hedge against the possibility that rates could increase and bond prices fall, and accordingly decided to set a floor on their portfolio by buying the CME put on the June five‐year T‐note with an exercise price of 120 and trading at 0‐59 or $921.875 per contract. The most likely‐to‐deliver bond on the underlying futures was the 1³/₈% T‐note maturing on 8/31/20; this bond had a duration of 4.02. Using the price‐sensitivity model, the managers would have needed to purchase 127 June T‐note futures puts with an exercise price of 120 for a cost of $117,078:

where

DurS = duration of the bond fund = 5.31

Durf = duration of the cheapest‐to‐deliver bond = 4.02

V0 = value of bond portfolio = $11,637,934

X = 120

Put premium per contract = $921.875

Future conversion factor on cheapest‐to‐deliver bond = CFA = 0.8371

T = time to expiration on the option as proportion of year = 141/365

YTMS = yield to maturity on the bond fund = 2.78%

YTMf = yield to maturity implied on the futures contract = 1.76%

Exhibit 8.13 shows:

1. The portfolio values and accrued interest on the option’s expiration of 5/20/16 given different interest rate shifts ranging from 50 basis point decreases to 50 basis point increases, and the portfolio’s corresponding yields to maturity and values.
2. The prices of the most‐likely‐to‐deliver bond (1³/₈% 8/31/20) on 5/20/16 given the different interest rate shifts and corresponding Treasury yields.
3. The corresponding June futures prices, where the futures prices are equal to the prices of the cheapest‐to‐deliver 1³/₈% T‐note plus its accrued interest (AI) divided by the underlying futures contracts conversion factor of 0.8371 (f_T = (S_T + AI)/CFA).
4. The cash flow from closing the 127 June 120 futures put options on the June futures expiration.
5. The put hedged value: the portfolio value plus accrued interest and cash flows from selling the puts at their IV.

Portfolio: Value on 3/15/16 = $11,637,934, Duration = 5.31, YTM = 2.78% Cheapest‐to‐Deliver T‐Note: 1 3/8 8/31/20: On 5/20/16 (Options Expiration), Accrued Interest (AI) = $0.303 per $100 Face Futures Price at T: (Cheapest‐to‐Deliver T‐Note Price + AI)/Conversion Factor = (ST + AI)/CFA Put Options on June 5‐Year T‐Note Futures with X = 120, Expiration = 5/20/15, Premium on 3/15/16 = ((59/64)/100)($100,000) = $921.875 Hedge: np = 127 Puts; Cost = (127)($921.875) = $117,078 Call Options on June 5‐Year T‐Note Futures with X = 120, Expiration = 5/20/15, Premium on 3/15/16 = ((59/64)/100)($100,000) = $921.875 Call Enhanced Portfolio: nC = 40 Call; Cost = (40)($921.875) = $36,875
1	2	3	4	5	6	7	8	9	10
Interest Rate Shift (Basis Points)	YTM	Portfolio Value	Accrued Interest	Portfolio Value + Accrued Interest	T‐Note YTM	T‐Note Price	T‐Note Accrued Interest	Futures Prices at T, fT = (ST + AI)/(CFA); CFA = 0.8317	Put Value PT = IV = Max[120 − fT,0]
50	2.96%	$11,699,113	$90,354	$11,789,467	1.8070%	$98.228	$0.303	118.47	$1.53
40	2.89%	$11,763,539	$90,354	$11,853,893	1.7070%	$98.637	$0.303	118.96	$1.04
30	2.76%	$11,828,553	$90,354	$11,918,907	1.6070%	$99.043	$0.303	119.45	$0.55
20	2.66%	$11,894,064	$90,354	$11,984,418	1.5070%	$99.454	$0.303	119.94	$0.06
10	2.56%	$11,960,925	$90,354	$12,051,279	1.4070%	$99.867	$0.303	120.44	$0.00
0	2.46%	$12,027,519	$90,354	$12,117,873	1.3070%	$100.585	$0.303	121.30	$0.00
−10	2.36%	$12,094,783	$90,354	$12,185,137	1.2070%	$100.698	$0.303	121.44	$0.00
−20	2.26%	$12,163,085	$90,354	$12,253,439	1.1070%	$101.117	$0.303	121.94	$0.00
−30	2.16%	$12,232,131	$90,354	$12,322,485	1.0070%	$101.537	$0.303	122.45	$0.00
−40	2.06%	$12,119,786	$90,354	$12,210,140	0.9070%	$101.960	$0.303	122.96	$0.00
−50	1.96%	$12,188,950	$90,354	$12,279,304	0.8070%	$102.384	$0.303	123.47	$0.00
	11	12	13	14	15	16	17	18
Interest Rate Shift (Basis Points)	Cash Flow from Selling 127 June Futures Puts at IV: (127) (IV/100) ($100,000)	Futures Put Hedge: Value of Portfolio plus Accrued Interest plus Cash Flow from Futures Puts: (5) + (11)	Hedged Portfolio Value Saved: (5) − (12)	Call Value CT = IV = Max[fT−120,0]	Cash Flow from Selling 40 June Futures Calls at IV: (40) (IV/100) ($100,000)	Futures Call Enhanced Portfolio: Value of Portfolio plus Accrued Interest plus Cash Flow from Futures Calls: (5) + (15)	Proportional Change in the Enhanced Portfolio: [Col (16)/$11,637,934]−1	Proportional Change in the Portfolio: [Col(5)/$11,637,934]−1
50	$194,386.20	$11,983,853	$194,386	$0.00	$0.00	$11,789,467	0.01302	0.01302
40	$131,932.19	$11,985,825	$131,932	$0.00	$0.00	$11,853,893	0.01856	0.01856
30	$69,936.28	$11,988,844	$69,936	$0.00	$0.00	$11,918,907	0.02414	0.02414
20	$7,176.87	$11,991,595	$7,177	$0.00	$0.00	$11,984,418	0.02977	0.02977
10	$0.00	$12,051,279	$0	$0.44	$55,887.94	$12,107,167	0.04032	0.03552
0	$0.00	$12,117,873	$0	$1.30	$165,526.03	$12,283,399	0.05546	0.04124
−10	$0.00	$12,185,137	$0	$1.44	$182,781.05	$12,367,918	0.06272	0.04702
−20	$0.00	$12,253,439	$0	$1.94	$246,762.05	$12,500,201	0.07409	0.05289
−30	$0.00	$12,322,485	$0	$2.45	$310,895.76	$12,633,381	0.08553	0.05882
−40	$0.00	$12,210,140	$0	$2.96	$375,487.56	$12,585,628	0.08143	0.04917
−50	$0.00	$12,279,304	$0	$3.47	$440,232.05	$12,719,536	0.09294	0.05511

As shown in the Exhibit 8.13, the hedge creates a floor for the portfolio value of around $11.9 million if rates increase by 10 basis point or more, while the portfolio gains in value if interest rates increase.

It should be noted that for a 50 basis points increase in rates, the puts provide a portfolio savings of $194,388 (= Hedge value − Unhedged value = $11,983,853 − $11,789,467), while for a 20 bp increase, the savings is only $7,177 (= Hedge value − Unhedged value = $11,991,595 − $11,984,418). With the cost of the insurance at $117,078, the managers may have concluded that this protection was not large enough, given the costs of the puts. Alternatively, they may have found a better hedging strategy would be to set up a range forward contract by selling a call with a higher exercise price to defray part of the cost of the puts.

Changing a Bond Portfolio’s Duration

In Chapter 4, we showed how the Xavier Bond Fund increased the fund’s duration from 5.43 to 6.08 in anticipation of a decrease in interest rates by going long in 16 June T‐note contract. Instead of using futures to change its portfolio’s exposure to interest rate, the managers could have alternatively used futures options. A bond portfolio manager who is very confident of an interest rate decrease (increase) could increase (decrease) her bond portfolio’s exposure by increasing (decreasing) the portfolio’s duration, Dur₀, to a new target duration, Dur_TR, by going long (short) in a bond futures contracts. The manager also could increase (decrease) her portfolio’s exposure by buying futures calls (puts). The number of option contracts needed to move the portfolio duration from Dur₀ to Dur_TR can be determined using the price‐sensitivity model in which:

where

• if Dur_TR > Dur₀, long in Call Options
• if Dur_TR < Dur₀, Long in Put Options

Changing interest rate exposure with options instead of futures provides an asymmetrical gain and loss exposure—a convex duration. For example, when calls (puts) are purchased to increase (decrease) the target duration, the option‐adjusted portfolio has a β_TR for interest rate decreases (decreases) and Dur₀ for interest rate increases (decreases). The cost of obtaining this asymmetrical relation is the cost of the options.

For example, suppose on 3/15/15, the Xavier Bond Fund manager strongly believed that rates would decrease and bond prices would rise, and as a result decided to use call options to increase the bond portfolio’s duration from 5.31 to 7.00. On 3/15/16, the CME call on the June five‐year T‐note with an exercise price of 120 was trading at 0‐59 or $921.875 per contract. (The most likely‐to‐deliver bond on the futures was the 1³/₈% T‐note maturing on 8/31/20; this bond had a duration of 4.02.) Using the price‐sensitivity model, the managers would have needed to purchase 40 June T‐note futures calls with an exercise price of 120 for a cost of $36,875:

As shown in Columns 14 to 18 in Exhibit 8.13, for downward yield curve shifts the call‐enhanced portfolio has a significant greater proportional returns than the unadjusted portfolio, reflecting a greater duration, while for upward shift the returns are the same with the calls out of the money.

Using Options to Set a Cap or Floor on a Cash Flow

In Chapter 4, we examined how a series or strip of Eurodollar futures contracts could be used to create a fixed or floating rate on the cash flow of an asset or liability. When there is a series of cash flows, such as a floating‐rate loan or an investment in a floating‐rate note, a strip of interest rate options can similarly be used to place a cap or a floor on the cash flow. For example, a company with a one‐year floating‐rate loan starting in September at a specified rate and then reset in December, March, and June to equal the spot LIBOR plus BP, could obtain a cap on the loan by buying a series of Eurodollar futures puts expiring in December, March, and June. At each reset date, if the LIBOR exceeds the discount yield on the put, the higher LIBOR applied to the loan will be offset by a profit on the nearest expiring put, with the profit increasing the greater the LIBOR; if the LIBOR is equal to or less than the discount yield on the put, the lower LIBOR applied to the loan will only be offset by the limited cost of the put. Thus, a strip of Eurodollar futures puts used to hedge a floating‐rate loan places a ceiling on the effective rate paid on the loan.

In the case of a floating‐rate investment, such as a floating‐rate note tied to the LIBOR or a bank’s floating rate loan portfolio, a minimum rate or floor can be obtained by buying a series of Eurodollar futures calls, with each call having an expiration near the reset date on the investment. If rates decrease, the lower investment return will be offset by profits on the calls; if rates increase, the only offset will be the limited cost of the calls.

Setting a Cap on a Floating‐Rate Loan with a Series of Eurodollar Puts

As an example of a cap, suppose Northwestern Bank offers Ryan’s Department Store a $20 million floating‐rate loan to finance its inventory over the next two years. The loan has a maturity of two years, starts on December 20, and is reset the next seven quarters. The initial quarterly rate on the loan is equal to 2%/4 (the current LIBOR of 1% plus 100 bp), the other rates are set on the quarterly reset dates equal to one fourth of the annual LIBOR on those dates plus 100 basis points: (LIBOR % + 1%)/4. Suppose Ryan’s wants to cap the loan by purchasing a strip of CME Eurodollar futures puts consisting of the seven put futures options. The top panel in Exhibit 8.14 shows seven Eurodollar futures puts each with an exercise price of $995,000 (IMM X = 98; R_D = 2) and with expirations coinciding with the reset dates on the loan.

T X P0 Cost of 20 Puts: 20($250)(P0)	3/20/T1 $995,000 2 $10,000	6/20/T1 $995,000 2.1 $10,500	9/20/T1 $995,000 2.2 $11,000	12/20/T1 $995,000 2.3 $11,500	3/20/T2 $995,000 2.4 $12,000	6/20/T2 $995,000 2.5 $12,500	9/20/T2 $995,000 2.6 $13,000	Total Cost = $80,500
1	2	3	4	5	6	7	8	9
Date	LIBOR %	Futures and Spot Price ST = fT	Put Cash Flow at Option’s Expiration 20(Max[$995,000 − fT,0]	Value of Put Cash Flow at Payment Date (Put CF at T)(1+LIBOR)0.25	Quarterly Interest at Payment Date 0.25 [(LIBOR + 0.01)] ($20m)	Hedged Debt Col 6 − Col 5	Hedged Rate [(4)(Col 7)]/$20m	Unhedged Rate LIBOR + 100bp
12/20	1.00
3/20	1.00	$997,500	$0		$100,000	$100,000.00	0.020	0.020
6/20	1.50	$996,250	$0	$0.00	$100,000	$100,000.00	0.020	0.020
9/20	2.00	$995,000	$0	$0.00	$125,000	$125,000.00	0.025	0.025
12/20	2.50	$993,750	$25,000	$0.00	$150,000	$150,000.00	0.030	0.030
3/20	3.00	$992,500	$50,000	$25,154.81	$175,000	$149,845.19	0.030	0.035
6/20	3.50	$991,250	$75,000	$50,370.85	$200,000	$149,629.15	0.030	0.040
9/20	4.00	$990,000	$100,000	$75,647.81	$225,000	$149,352.19	0.030	0.045
12/20				$100,985.34	$250,000	$149,014.66	0.030	0.050

Expiring futures (or settlement price):

Given these Eurodollar futures puts, Ryan’s could cap the floating‐rate loan by buying a strip of 20 Eurodollar futures puts for a total cost of $80,500. The lower panel in Exhibit 8.14 shows the put‐hedged rates on the loan and the unhedged rates for an increasing interest rate scenario in which the LIBOR increases from 1% on March 20 to 4% seven quarters later. The numbers in the exhibit show for each period, Ryan’s quarterly interest payments, option cash flow, option values at the interest payment date, hedged interest payments (interest minus option cash flow), and hedged rate as a proportion of a $20 million loan. The scenarios shown assume that the options’ expiration dates coincide with the reset dates. As shown, the put options allow Ryan’s to cap its loan at 3.00%, when the LIBOR is greater than 2%, while benefiting with lower rates on its loan when the LIBOR is less than 2%.

Setting a Floor on a Floating‐Rate Investment with a Series of Eurodollar Calls

As a an example of a floor, suppose Kendall Trust is planning to invest $20 million in a Northwestern Bank two‐year floating‐rate note paying LIBOR plus 100 basis points. The investment starts on 12/20 at 2% (when the LIBOR = 1%) and is then reset the next seven quarters. Suppose Kendall Trust would like to establish a floor on the rates it obtains on the floating‐rate note with a strip of the seven CME Eurodollar futures call options shown in the top panel in Exhibit 8.15. Each of the Eurodollar futures calls has an exercise price of $995,000 (IMM X = 98; R_D = 2) and expirations coinciding with the reset dates on the floating‐rate note.

T	3/20/T1	6/20/T1	9/20/T1	12/20/T1	3/20/T2	6/20/T2	9/20/T2
X	$995,000	$995,000	$995,000	$995,000	$995,000	$995,000	$995,000
C0	2	2.1	2.2	2.3	2.4	2.5	2.6
Cost of 20 Calls: 20($250)(C0)	$10,000	$10,500	$11,000	$11,500	$12,000	$12,500	$13,000	Total Cost = $80,500
1	2	3	4	5	6	7	8	9
Date	LIBOR %	Futures and Spot Price ST = fT	Call Cash Flow at Option’s Expiration 20(Max[fT − $995,000,0]	Value of Call Cash Flow at Receipt Date (Call CF at T)(1+LIBOR)0.25	Quarterly Interest at Receipt Date 0.25 [(LIBOR + 0.01)] ($20m)	Hedged Interest Revenue Col 6 + Col 5	Hedged Rate [(4)(Col 7)]/$20m	Unhedged Rate LIBOR + 100bp
12/20	1.00
3/20	1.00	$997,500	$50,000		$100,000	$100,000.00	0.020	0.020
6/20	1.50	$996,250	$25,000	$50,124.53	$100,000	$150,124.53	0.030	0.020
9/20	2.00	$995,000	$0	$25,093.23	$125,000	$150,093.23	0.030	0.025
12/20	2.50	$993,750	$0	$0.00	$150,000	$150,000.00	0.030	0.030
3/20	3.00	$992,500	$0	$0.00	$175,000	$175,000.00	0.035	0.035
6/20	3.50	$991,250	$0	$0.00	$200,000	$200,000.00	0.040	0.040
9/20	4.00	$990,000	$0	$0.00	$225,000	$225,000.00	0.045	0.045
12/20				$0.00	$250,000	$250,000.00	0.050	0.050

Expiring futures (or settlement price):

To set the floor on the floating note, Kendall would need to buy 20 Eurodollar call strips for a total cost of $80,500. The lower panel of Exhibit 8.15 shows Kendall Trust’s quarterly interest receipts, option cash flow, option values at the interest payment dates, hedged interest revenue (interest plus option cash flow), and hedged rate as a proportion of a $20 million investment for each period with the assumption that the reset dates and option expiration dates coincide. The assumed LIBOR rate shown in the exhibit reflect an increasing interest rate scenario in which the LIBOR increases from 1% on March 20 to 4% seven quarters later. As shown, the call options allow Kendall to attain a floor on investment rate of 3% when the LIBOR is 2% or less with the benefit of higher yields when the LIBOR is greater than 3%.

Setting a Cap on a Floating‐Rate Loan with a Series of Caplets—Cap

As noted in Chapter 6, a popular option offered by financial institutions is the cap: a series of European interest rate call options—a portfolio of caplets. For example, a 2.5%, two‐year cap on a three‐month LIBOR, with a NP of $100 million, provides, for the next two years, a payoff every three months of (LIBOR − 0.025)(0.25)($100,000,000) if the LIBOR on the reset date exceeds 2.5% and nothing if the LIBOR equals or is less than 2%. Caps are often written by financial institutions in conjunction with a floating‐rate loan and are used by buyers as a hedge against interest rate risk.

As an example, suppose the Jones Development Company borrows $100 million from Southern Bank to finance a two‐year construction project. Suppose the loan is for two years, starting on March 1 at a known rate of 2.5% (1% LIBOR plus 150 bp) then resets every three months—6/1, 9/1, 12/1, and 3/1—at the prevailing LIBOR plus 150 BP. In entering this loan agreement, suppose the Jones Company is uncertain of future interest rates and therefore would like to lock in a maximum rate, while still benefiting from lower rates if the LIBOR decreases. To achieve this, suppose the Jones Company buys a cap corresponding to its loan from Southern Bank for $150,000, with the following terms:

• The cap consist of seven caplets with the first expiring on 6/1/Y1 and the others coinciding with the loan’s reset dates
• Exercise rate on each caplet = 2.5%
• NP on each caplet = $100 million
• Reference rate = LIBOR
• Time period to apply to the payoff on each caplet = 90/360
• Payment date on each caplet is at the loan’s interest payment date, 90 days after the reset date
• The cost of the cap = $150,000

On each reset date, the payoff on the corresponding caplet would be

With the 2.5% exercise rate or cap rate, the Jones Company would be able to lock in a maximum rate each quarter equal to the cap rate plus the basis points on the loan (4%), while still benefiting with lower interest costs if rates decrease. This can be seen in Exhibit 8.16, where the quarterly interests on the loan, the cap payoffs, and the hedged and unhedged rates are shown for different assumed LIBORs at each reset date on the loan. For the four reset dates from 6/1/Y2 to the end of the loan, the LIBOR exceeds 2.5%. In each of these cases, the higher interest on the loan is offset by the payoff on the cap yielding a hedged rate on the loan of 4.0% (the 4% rate excludes the $150,000 cost of the cap). For the first two reset dates on the loan, 6/1/Y1, 9/1/Y1, and 12/1/Y1, the LIBOR is less than the cap rate. At these rates, there is no payoff on the cap, but the rates on the loan are lower with the lower LIBORs.

Loan: Floating Rate Loan; Term = 2 years; Reset Dates: 6/1, 9/1, 12/1; Time Frequency = 0.25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date Cap: Cost of Cap = $150,000; Cap Rate = 2.5%; Reference Rate = LIBOR; Time Frequency = 0.25; Caplets’ Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date
1	2	3	4	5	6	7
Reset Date	Assumed LIBOR	Loan Interest on Payment Date (LIBOR + 150bp)(0.25)($100m)	Cap Payoff on Payment Date Max[LIBOR − 0.025,0](0.25)($100m)	Hedged Interest Payment Col. (3) − Col. (4)	Hedged Rate 4[Col (5)/$100m]	Unhedged Rate LIBOR + 150bp
3/1/Y1n	0.010
6/1/Y1	0.010	$625,000	$0	$625,000	0.025	0.025
9/1/Y1	0.015	$625,000	$0	$625,000	0.025	0.025
12/1/Y1	0.020	$750,000	$0	$750,000	0.030	0.030
3/1/Y2	0.025	$875,000	$0	$875,000	0.035	0.035
6/1/Y2	0.030	$1,000,000	$0	$1,000,000	0.040	0.040
9/1/Y2	0.035	$1,125,000	$125,000	$1,000,000	0.040	0.045
12/1/Y2	0.040	$1,250,000	$250,000	$1,000,000	0.040	0.050
3/1/Y3		$1,375,000	$375,000	$1,000,000	0.040	0.055

ⁿ There is no cap on this date

Asset: Floating rate loan made by bank; Term = 2 years; Reset Dates: 3/1, 6/1, 9/1, 12/1; Time Frequency = 0.25; Rate = LIBOR + 150BP; Payment Date = 90 days after reset date Floor: Cost of Floor = $100,000; Floor Rate = 2.5%; Reference Rate = LIBOR; Time Frequency = .25; Floorlets’ Expirations: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date
1	2	3	4	5	6	7
Reset Date	Assumed LIBOR	Interest Received on Payment Date (LIBOR + 150bp)(0.25)($100m)	Floor Payoff on Payment Date Max[0.025−LIBOR,0](0.25)($100m)	Hedged Interest Income Col. (3) + Col. (4)	Hedged Rate 4[Col (5)/$100M]	Unhedged Rate LIBOR + 150bp
3/1/Y1n	0.010
6/1/Y1	0.010	$625,000	$0	$625,000	0.025	0.025
9/1/Y1	0.015	$625,000	$375,000	$1,000,000	0.040	0.025
12/1/Y1	0.020	$750,000	$250,000	$1,000,000	0.040	0.030
3/1/Y2	0.025	$875,000	$125,000	$1,000,000	0.040	0.035
6/1/Y2	0.030	$1,000,000	$0	$1,000,000	0.040	0.040
9/1/Y2	0.035	$1,125,000	$0	$1,125,000	0.045	0.045
12/1/Y2	0.040	$1,250,000	$0	$1,250,000	0.050	0.050
3/1/Y3		$1,375,000	$0	$1,375,000	0.055	0.055

ⁿ There is no floor on this date

Setting a Floor on a Floating‐Rate investment with a Series of Floorlets—Floor

As explained in Chapter 6, a plain‐vanilla floor is a series of European interest rate put options—a portfolio of floorlets. For example, a 2.5%, two‐year floor on a three‐month LIBOR, with a NP of $100 million, provides, for the next two years, a payoff every three months of (0.025 − LIBOR)(0.25)($100,000,000) if the LIBOR on the reset date is less than 2.5% and nothing if the LIBOR equals or exceeds 2.5%. Floors are often purchased by investors as a tool to hedge their floating‐rate investments against interest rate declines. Thus, with a floor, an investor with a floating‐rate security is able to lock in a minimum rate each period, while still benefiting from higher yields if rates increase.

As an example, suppose Southern Bank in the above example wanted to establish a minimum rate or floor on the rates it was to receive on the two‐year floating‐rate loan it made to the Jones Development Company. To this end, suppose the bank purchased from another financial institution a floor for $100,000 with the following terms corresponding to its floating‐rate asset:

• The floor consist of seven floorlets with the first expiring on 6/1/Y1 and the others coinciding with the reset dates on the bank’s floating‐rate loan to the Jones Company
• Exercise rate on each floorlet = 2.5%
• NP on each floorlet = $100 million
• Reference rate = LIBOR
• Time period to apply to the payoff on each floorlet = 90/360
• The cost of the floor = $100,000; it is paid at beginning of the loan, 3/1/Y1

On each reset date, the payoff on the corresponding floorlet would be

With the 2.5% exercise rate, Southern Bank would be able to lock in a minimum rate each quarter equal to the floor rate plus the basis points on the floating‐rate asset (4%), while still benefiting with higher returns if rates increase. In Exhibit 8.17, Southern Bank’s quarterly interests received on its loan to Jones, its floor payoffs, and its hedged and unhedged yields on its loan asset are shown for different assumed LIBORs at each reset date. For the first three reset dates on Southern Bank’s loan to Jones, 6/1/Y1, 9/1/Y1, and 12/1/Y1, the LIBOR is less than the floor rate of 2.5%. At these rates, there is a payoff on the floor that compensates Southern Bank for the lower interest it receives on the loan; this results in a hedged rate of return on the bank’s loan asset of 4% (the cost of the floor excluded). For the five reset dates from 6/1/Y1 to the end of the loan, the LIBOR equals or exceeds the floor rate. At these rates, there is no payoff on the floor, but the rates the bank earns on its loan to Jones are greater, given the greater LIBORs.

Collars and Corridors

A collar is a combination of a long position in a cap and a short position in a floor with different exercise rates. The sale of the floor is used to defray the cost of the cap. For example, the Jones Development Company in our previous case could reduce the cost of the cap it purchased to hedge its floating‐rate rate loan by selling a floor. By forming a collar to hedge its floating‐rate debt, the Jones Company, for a lower net hedging cost, would still have protection against a rate movement against the cap rate, but it would have to give up potential interest savings from rate decreases below the floor rate. For example, suppose the Jones Company decided to defray the $150,000 cost of its 2.5% cap by selling a 1% floor for $75,000, with the floor having similar terms to the cap (effective dates on floorlet = reset dates, reference rate = LIBOR, NP on floorlets = $100 million, and time period for rates = 0.25). By using the collar instead of the cap, Jones reduces its hedging cost from $150,000 to $75,000, and the company can still lock in a maximum rate on its loan of 4%. However, when the LIBOR is less than 1%, the company has to pay on the 1% floor, offsetting the lower interest costs it would pay on its loan. See Exhibit 8.18.

Loan: Floating Rate Loan; Term = 2 years; Reset Dates: 3/1, 6/1, 9/1, 12/1; Time Frequency = 0.25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date Cap Purchase: Cost of Cap = $150,000; Cap Rate = 2.5%; Reference Rate = LIBOR; Time Frequency = 0.25; Caplets’ Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date. Floor Sale: Sale of Floor = $75,000; Floor Rate = 1%; Reference Rate = LIBOR; Time Frequency = 0.25; Floorlets’ Expiration: On loan reset dates, starting at 61/Y1; Payoff Date = 90 days after reset date.
1	2	3	4	5	6	7	8
Reset Date	Assumed LIBOR	Loan Interest (LIBOR + 150bp)(.25)($100M)	Cap Payoff Max[LIBOR − 0.025,0](0.25)($100m)	Floor Payment Max[0.01 − LIBOR,0](0.25)($100m)	Hedged Interest Payment Col. (3) − Col. (4) + Col (5)	Hedged Rate 4[Col (6)/$100m]	Unhedged Rate LIBOR + 150bp
3/1/Y1n	0.010
6/1/Y1	0.005	$625,000	$0	$0	$625,000	0.025	0.025
9/1/Y1	0.0075	$500,000	$0	$125,000	$625,000	0.025	0.020
12/1/Y1	0.010	$562,500	$0	$62,500	$625,000	0.025	0.0225
3/1/Y2	0.020	$625,000	$0	$0	$625,000	0.025	0.025
6/1/Y2	0.0225	$875,000	$0	$0	$875,000	0.035	0.035
9/1/Y2	0.030	$937,500	$0	$0	$937,500	0.0375	0.0375
12/1/Y2	0.035	$1,125,000	$125,000	$0	$1,000,000	0.040	0.045
3/1/Y3		$1,250,000	$250,000	$0	$1,000,000	0.040	0.050

ⁿ Loan interest, cap payoff, and floor payment made on payment date

An alternative financial structure to a collar is a corridor. A corridor is a long position in a cap and a short position in a similar cap with a higher exercise rate. The sale of the higher exercise‐rate cap is used to partially offset the cost of purchasing the cap with the lower strike rate. For example, the Jones Company, instead of selling a 1% floor for $75,000 to partially finance the $150,000 cost of its 2.5% cap, could sell a 3.5% cap for, say, $75,000. If cap purchasers, however, believe there was a greater chance of rates increasing than decreasing, they would prefer the collar to the corridor as a tool for financing the cap. In practice, collars are more frequently used than corridors.

A reverse collar is a combination of a long position in a floor and a short position in a cap with different exercise rates. The sale of the cap is used to defray the cost of the floor. For example, suppose that Southern Bank in our above floor example decided to reduce the $100,000 cost of the 2.5% floor it purchased to hedge the floating‐rate loan it made to the Jones Company by selling a 4% cap for $50,000, with the cap having similar terms to the floor. By using the reverse collar instead of the floor, Southern Bank reduces its hedging cost from $100,000 to $50,000, and the bank can still lock in a minimum rate on its investment of 2.5%. However, when the LIBOR is greater than 4%, the bank rates on the investment are fixed at 5.5%, offsetting the higher interest it would have received. See Exhibit 8.19. Finally, note that instead of financing a floor with a cap, an investor could form a reverse corridor by selling another floor with a lower exercise rate.

Asset: Floating rate loan made by bank; Term = 2 years; Reset Dates: 3/1, 6/1, 9/1, 12/1; Time Frequency = 0.25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date Floor Purchase: Cost of Floor = $100,000; Floor Rate = 2.5%; Reference Rate = LIBOR; Time Frequency = 0.25; Floorlets’ Expirations: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date. Cap Sale: Revenue from Cap = $50,000; Cap Rate = 4%; Reference Rate = LIBOR; Time Frequency = 0.25; Caplets’ Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
1	2	3	4	5	6	7	8
Reset Date	Assumed LIBOR	Interest Received (LIBOR + 150bp)(0.25) ($100m)	Floor Payoff Max[0.025−LIBOR,0] (0.25)($100m)	Cap Payment Max[LIBOR−0.04,0] (0.25)($100m)	Hedged Interest Income Col. (3) + Col. (4) − Col (5)	Hedged Rate 4[Col (5)/$100M]	Unhedged Rate LIBOR + 150bp
3/1/Y1n	0.01000
6/1/Y1	0.01500	$625,000	$0	$0	$625,000	0.025	0.025
9/1/Y1	0.02000	$750,000	$250,000	$0	$1,000,000	0.040	0.030
12/1/Y1	0.02750	$875,000	$125,000	$0	$1,000,000	0.040	0.035
3/1/Y2	0.03000	$1,062,500	$0	$0	$1,062,500	0.0425	0.043
6/1/Y2	0.04000	$1,125,000	$0	$0	$1,125,000	0.045	0.045
9/1/Y2	0.04500	$1,375,000	$0	$0	$1,375,000	0.055	0.055
12/1/Y2	0.05000	$1,500,000	$0	$125,000	$1,375,000	0.055	0.060
3/1/Y3		$1,625,000	$0	$250,000	$1,375,000	0.055	0.065

ⁿ Loan interest, floor payoff, and cap payment made on payment date

Conclusion

Stock, index, currency, commodity, and bond futures and spot options are valuable tools in managing security and portfolio positions. These derivative contracts make it possible for stock and fixed‐income portfolio managers to obtain portfolio insurance or to cap the cost of buying a security in the future. Managers also can use derivatives to adjust the beta of a portfolio if they anticipate a bull or bear market, or the duration of a fixed‐income portfolio, if they are anticipating an interest rate increase or decrease, thus eliminating the need to reallocate their portfolio allocation. Finally, futures options on commodities and currencies provide a tool for setting caps and floors for commodity and currency purchases and sale.

In our examination of option strategies and hedging positions in this and the last chapter, we have assumed a price for the option. As we examined earlier with futures pricing, the pricing of options is based on arbitrage. In Part 3, we examine option pricing models.

Selected References

1. Clarke, R., and R. Arnott. “The Cost of Portfolio Insurance: Tradeoffs and Choices.” Financial Analysts Journal 43 (November–December 1987): 35–47.
2. Etzioni, E. “Rebalance Disciplines for Portfolio Insurance.” Journal of Portfolio Management 13 (Fall 1986): 59–62.
3. Figlewski, S. “Hedging Performance and Basis Risk in Stock Index Futures.” Journal of Finance 39 (July 1984): 657–669.
4. Figlewski, S., and S. Kin. “Portfolio Management with Stock Index Futures.” Financial Analysts Journal 38 (January‐February 1982): 52–60.
5. Grant, D. “How to Optimize with Stock Index Futures.” Journal of Portfolio Management 8 (Spring 1982): 32–36.
6. Gressis, N., G. Glahos, and G. Philippatos. “A CAPM‐Based Analysis of Stock Index Futures.” Journal of Portfolio Management 10 (Spring 1984): 47–52.
7. Kolb, R. W., and R. Chiang. “Improving Hedging Performance Using Interest Rate Futures.” Financial Management 10 (Fall 1981): 72–79.
8. Leland, H. “Who Should Buy Portfolio Insurance?” Journal of Finance 35 (May 1980): 581–594.
9. Madura, J., and T. Veit. “Use of Currency Options in International Cash Management.” Journal of Cash Management (January‐February 1986): 42–48.
10. Madura, J., and E. Nosari. “Utilizing Currency Portfolios to Mitigate Exchange Rate Risk.” Columbia Journal of World Business (Spring 1984): 96–99.
11. McCable, G., and C. Franckle. “The Effectiveness of Rolling the Hedge Forward in the Treasury Bill Futures Market.” Financial Management 12 (Summer 1983): 21–29.
12. O’Brien, T. “The Mechanics of Portfolio Insurance?” Journal of Portfolio Management (Spring 1988): 40–47.
13. Pozen, R. “The Purchase of Protective Puts by Financial Institutions.” Financial Analysts Journal 34 (July/August 1978): 47–60.
14. Rendleman, R., and C. Carabini. “The Efficiency of the Treasury Bill Futures Market.” Journal of Finance 34 (September 1979): 895–914.
15. Resnick, B., and E. Hennigar. “The Relation Between Futures and Cash Prices for US Treasury Bonds.” Review of Research in Futures Markets 2 (1983): 282–299.
16. Senchak, A., and J. Easterwood. “Cross Hedging CD’s with Treasury Bill Futures.” Journal of Futures Markets 3 (1983): 429–438.
17. Siegel, D., and D. Siegel. Futures Markets. Chicago: Dryden Press, 1990, 203–342 and 493–504.
18. Stokes, H., and H. Neuburger. “Interest Arbitrage, Forward Speculation and the Determination of the Forward Exchange Rate.” Columbia Journal of World Business 4 (1979): 86–99.
19. Swanson, P., and S. Caples. “Hedging Foreign Exchange Risk Using Forward Exchange Markets: An Extension.” Journal of International Business Studies (Spring 1987): 75–82.
20. Toevs, A., and D. Jacob. “Futures and Alternative Hedge Methodologies.” Journal of Portfolio Management (Spring 1986): 60–70.
21. Viet, T., and W. Reiff. “Commercial Banks and Interest Rate Futures: A Hedging Survey.” Journal of Futures Markets 3 (1983): 283–293.
22. Virnola, A., and C. Dale. “The Efficiency of the Treasury Bill Futures Market: An Analysis of Alternative Specifications.” Journal of Financial Research 3 (1980): 169–188.

Problems and Questions

Note: A number of problems can be done in Excel.

1. The Bryce Investment Trust Company plans to liquidate part of its stock portfolio in June. The company is bullish but would like to set a floor on the portfolio sale as a defensive strategy. The portfolio it plans to sell is well diversified, has a beta of 1.5, and is currently worth $100 million. The S&P 500 index is currently at 2,250 and a June S&P 500 spot put option with an exercise price of 2,250 and multiplier of $100 is currently at 50.
   1. Using the price‐sensitivity model, determine how many S&P 500 spot put contracts the Bryce Investment Trust Company would need in order to set a floor on the sale of its $100 million portfolio in June. What is the cost of the puts?
   2. Show in a table the proportional changes in the S&P 500 from its current level of 2,250, the proportional changes in the portfolio, the values of the portfolio corresponding to the spot index, the put option values, the put position’s cash flow, and the put hedged portfolio values on the June expiration date for possible spot index values of index values for possible spot index values starting at 1,980 with 30 point steps to 2,850.
2. Suppose the Bryce Investment Trust Company in Problem 1 would like to defray part of the cost of the portfolio put insurance by setting up a short‐range forward contract by selling a June S&P 500 spot call option with an exercise price of 2,500, multiplier of $100, and premium of 35.
   1. Using the price‐sensitivity model, determine how many June S&P 500 call contracts the Bryce Investment Trust Company would need to sell to form a short‐range forward contract with its long June put position. What is the revenue from the calls and the net cost of the short‐range forward contract?
   2. Show in a table the proportional changes in the S&P 500 from its current level of 2,250, the proportional changes in the portfolio, the values of the portfolio corresponding to the spot index, the long put option values, the long put position’s cash flow, the short call option values, the short call position’s cash flow, and the short‐range forward portfolio values on the June expiration date for possible spot index values starting at 1,980 with 30 point steps to 2,850.
3. The Keynes Investment Company is expecting a $100 million inflow of cash in June and is planning to invest the cash in a portfolio of stocks with a β = 1.5. Keynes Investments is concerned there will be a strong bull market and would like to cap its June portfolio investment cost with June S&P 500 spot call options with an exercise price of 2,250, multiplier of $100, and premium of 35. Currently, the S&P 500 index is at 2,250.
   1. Using the price‐sensitivity model, determine how many S&P 500 spot call contracts the Keynes Investment Trust Company would need in order to set a $100 million cap on the purchase of its portfolio in June. What is the cost of the call?
   2. Show in a table the proportional changes in the S&P 500 from its current level of 2,250, the proportional changes in the portfolio, the values of the portfolio corresponding to the spot index, the call option values, the call position’s cash flow, and the call‐hedged portfolio cost on the June expiration date for possible spot index values of index values for possible spot index values starting at 1,900 with 25 point steps to 2,500.
4. Suppose the Keynes Investment Company in Problem 3 would like to defray part of the cost of its cap by setting up a long‐range forward contract with a short position in June S&P 500 spot put options with an exercise price of 2,000, multiplier of $100, and premium of 35.
   1. Using the price‐sensitivity model, determine how many June S&P 500 put contracts the Keynes Investment Company would need to sell to form a long‐range forward contract with its long June call position. What is the revenue from the put and the net cost of the long‐range forward contract?
   2. Show in a table the proportional changes in the S&P 500 from its current level of 2,250, the proportional changes in the portfolio, the values of the portfolio corresponding to the spot index, the long call option values, the long call position’s cash flows, the short put option values, the short put position’s cash flow, and the long‐range forward portfolio cost on the June expiration date for possible spot index values starting at 1,900 with 25 point steps to 2,500.
5. The Hunter Investment Company manages a well‐diversified equity fund. The current value of the portfolio it manages is $100 million and the portfolio has a beta of 1.0. Hunter Investment is bullish about the market and would like to increase its portfolio beta to 1.5 if the market is 2,250 or greater in June, while keeping its beta at one if the market is less than 2,250. Currently, the S&P 500 spot index is trading at 2,250 and a June S&P 500 spot call contract with an exercise price of 2,250 and $100 multiplier is trading at 50.
   1. Using the price‐sensitivity model, determine how many S&P 500 call contracts the Hunter Investment Company would need in order to increase its portfolio beta to 1.5 in June if the market is 2,250 or greater while keeping its portfolio at one if the market is 2,250 or less.
   2. Show in a table the proportional changes in the S&P 500 from its current level of 2,250, the proportional changes in the portfolio, the values of the portfolio corresponding to the spot index, the long call option values, the long call position’s cash flow, the call‐enhanced portfolio values, and the proportion changes in the call‐enhanced portfolio from its $100 million current value, proportion changes in the call‐enhanced portfolio value to the proportional changes in S&P 500. Evaluate for possible spot index values on the June expiration date starting at 2,000 with 25 point steps to 2,600.
   3. Comment on the convexity of the call‐enhanced portfolio’s beta.
6. Suppose ESPN (Disney) is expecting revenues of £6,250,000 next April (one year from now: T = 1) from its United Kingdom Rugby sports productions division. ESPN expects the dollar price of the British pound to increase when it converts its £6,250,000 but would like to set a floor on the dollar value of its revenue. Currently, the spot $/£ exchange rate is $1.50/£, and there is a put option on the April BP futures contract with an exercise price of $1.50/£, contract size of £62,500 trading at $0.02/£.
   1. Explain how ESPN could set a floor on it dollar revenue in June with the British pound futures put option. How many contracts would they need to set up the floor?
   2. Show in a table ESPN’s April dollar revenue from converting £6,250,000, the intrinsic values of the BP future put, the long put position’s cash flow, and the put‐insured dollar revenue on April expiration date for possible spot exchange rates of $1.00/£, $1.20/£, $1.50/£, $1.50/£, $1.60/£, $1.70/£, and $1.90/£. Assume the expiring futures price is equal to the spot $/£ exchange rate.
7. ESPN owes the United Kingdom Rugby Association £6,250,000 as payment for the exclusive licensing right to produce rugby matches on the European and US cable markets. ESPN expects the dollar price of the British pound to decrease when it buys £6,250,000 to pay for its obligation but would like to set a cap on the dollar cost of its British pounds. Currently, the spot $/£ exchange rate is $1.50/£, and there is a call option on the April BP futures contract with an exercise price of $1.50/BP and contract size of 62,500 BP trading at $0.02/£.
   1. Explain how ESPN could cap the dollar cost of its British pounds in April with the British pound futures call option. How many contracts would ESPN need to buy to set up the cap?
   2. Show in a table ESPN’s April dollar cost from purchasing £6,250,000, the intrinsic values of the BP futures call, the long call position’s cash flow, and the call‐capped dollar cost on the April expiration date for possible spot exchange rates of $1.00/£, $1.20/£, $1.50/£, $1.50/£, $1.60/£, $1.70/£, and $1.90/£. Assume the expiring futures price is equal to the spot $/£ exchange rate.
8. Ms. Hunter is the chief financial officer for Atlanta Developers. In January, she estimates that the company will need to purchase 300,000 square feet of plywood in June to meet its material needs on one of its office construction jobs. Ms. Hunter is confident that plywood prices will be decreasing in the future but does not want to assume the price risk if plywood prices were to increase. Suppose there is a June plywood futures call with X = $0.20/sq. ft. (contract size is 5,000 square feet), selling at C = $0.02, and expiring at the same time as the June underlying plywood futures contract. Explain how Ms. Hunter could cap the company’s June plywood costs with a position in the June plywood futures call. Evaluate the cap by showing in a table Ms. Hunter’s hedged costs at the plywood futures option’s expiration date by buying the plywood on the spot market and closing the futures call options at their intrinsic value. Evaluate at possible spot prices of 0.14/sq. ft., 0.16/sq. ft., 0.18/sq. ft., 0.20/sq. ft., 0.22/sq. ft., 0.24/sq. ft., 0.26/sq. ft., and 0.28/sq. ft.
9. In May, Mr. Smith planted a wheat crop that he expects to harvest in September. He anticipates the September harvest to be 100,000 bushels. While he expects wheat prices to increase, he would still like downside protection against any unexpected decrease in wheat prices. Suppose there is a September wheat futures put contract available with an exercise price of X = $4.40/bu. (contract size of 5,000 bushels), expiring at the same time as underlying September wheat futures, and currently trading at $0.05. Explain how Mr. Smith could obtain downside protection by buying the put. Show in a table Mr. Smith’s hedged revenue at the futures’ expiration date from closing the futures put at the intrinsic value and selling his 100,000 bushels of wheat on the spot market at possible spot prices of $3.60/bu., $3.80/bu., $4.00/bu., $4.20/bu., $4.40/bu., $4.60/bu., $4.80/bu., and $5.00/bu.
10. A fixed‐income fund manager plans to sell 20 $100,000 face value T‐bonds from her government fund in March. The T‐bonds she plans to sell pay 3% interest and are currently priced at 105. At the anticipated selling date, the bonds will have 15 years to maturity and no accrued interest. The manager believes that long‐term rates could decrease but does not want to risk selling the bonds at lower prices if rates increase. For $20,000, the manager can purchase an OTC T‐bond option on her bonds from a dealer at an exercise price equal the current price and expiration coinciding with her March T‐bond sales date.
    1. Describe the OTC option and its terms.
    2. Show in a table the manager’s option‐hedged revenue (do not include option cost) for possible spot T‐bond prices at the March sale of 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, and 110. Assume the manager will exercise her option, if it is feasible (instead of closing), and that she will sell her bonds in the market, if it is not feasible.
11. Suppose the fixed‐income fund manager in Question 10 were expecting a cash flow of $2,000,000 in March and planned to invest the cash flow in twenty T‐bonds each with a face value of $100,000. Suppose the T‐bonds she plans to buy pay 3% interest, have 15 years to maturity, and are currently priced at 105. At the anticipated purchase date, assume such bonds will have no accrued interest. The manager believes that long‐term rates could increase but does not want to risk buying the bond at higher prices if rates decrease. For $20,000, the manager can purchase an OTC T‐bond option on the 20 bonds from a dealer at an exercise price equal the current price and expiration coinciding with her March T‐bonds purchase date:
    1. Describe the OTC option and its terms.
    2. Show in a table the manager’s option‐hedged cost (do not include option cost) for possible spot T‐bond prices at the March purchase date of 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, and 110. Assume the manager will exercise her option, if it is feasible (instead of closing), and that she will buy her bonds in the market, if it is not feasible.
12. The O’Brien Beverage Company is considering capping a two‐year $10 million floating‐rate loan from First National Bank with a strip of Eurodollar puts. O’Brien’s floating‐rate loan starts on December 20 with the rate on the loan reset each quarter. The initial quarterly rate is equal to 3.5%/4, the other rates are set each quarter on 3/20, 6/20, 9/20, and 12/20 over the next seven quarters to equal one fourth of the annual LIBOR on those dates plus 100 basis points: (LIBOR % + 1%)/4. The top panel in Table 8.1 shows seven Eurodollar futures put contracts with expirations coinciding with O’Brien’s floating‐rate loan available and their premiums on December 20. Each futures put has an exercise price of $993,750 (X = 98.5 CME index).
    1. Explain how the O’Brien Beverage Company could attain a cap on its floating‐rate loan with the put options shown in Table 8.1. What is the cost of the put strip?
    2. Complete Table 8.1, showing the company’s quarterly interest payments, option cash flow, hedged interest payments (interest minus option cash flow), and hedged rate as a proportion of a $10 million loan (do not include option cost) given the LIBOR rates shown in the table.

    T	3/20/T1	6/20/T1	9/20/T1	12/20/T1	3/20/T2	6/20/T2	9/20/T2
    X	$993,750	$993,750	$993,750	$993,750	$993,750	$993,750	$993,750
    P0	4	4.2	4.4	4.6	4.8	5	5.2
    Cost of 10 Puts:								Total Cost =
    1	2	3	4	5	6	7	8	9
    Date	LIBOR %	Futures and Spot Price ST = fT	Put Cash Flow at Option’s Expiration 10(Max[$993,750 − fT,0]	Value of Put Cash Flow at Payment Date (Put CF at T)(1+LIBOR).25	Quarterly Interest at Payment Date [(LIBOR + 0.01)] ($:	Hedged Debt Col 6 − Col 5	Hedged Rate [(4)(Col 7)]/$10m	Unhedged Rate LIBOR + 100bp
    12/20	1.50
    3/20	1.50	$996,250	$0		$62,500	$62,500.00	0.025	0.025
    6/20	2.00	$995,000	$0	$0.00	$62,500	$62,500.00	0.025	0.025
    9/20	2.50
    12/20	3.00
    3/20	3.50
    6/20	4.00
    9/20	4.50
    12/20

13. Northern Trust is considering setting a floor on a two‐year $10 million investment in a floating‐rate note (FRN) from First National Bank with a strip of Eurodollar calls. The FRN pays LIBOR plus 100 basis points, starts on December 20 with the initial quarterly rate equal to 3.5%/4 and the other rates reset each quarter on 3/20, 6/20, 9/20, and 12/20 over the next seven quarters to equal to one fourth of the annual LIBOR on those dates plus 100 basis points: (LIBOR % + 1%)/4. The top panel in Table 8.2 shows on December 20 seven Eurodollar futures call contracts with expirations coinciding with First National Bank’s FRN and their premium on December 20. The exercise price on each call is $993,750 (X = 98.5 CME index).
    1. Explain how the Northern Trust could attain a floor on its floating‐rate note with the call options shown in Table 8.2. What is the cost of the call strip?
    2. Complete Table 8.2, showing Northern Trust’s quarterly interest receipt, option cash flow, hedged interest receipt (interest plus option cash flow), and hedged rate as a proportion of a $10 million investment (do not include option cost) for each period, given the LIBOR rates shown in the table.
14. Suppose Eastern Bank offers Gulf Refinery a $150 million floating‐rate loan along with a cap to finance the purchase of its drilling equipment. The floating‐rate loan has a maturity of two years, starts on December 20, and is reset the next seven quarters. The initial quarterly rate is equal to 2.5%/4 and the other rates are reset quarterly to equal to one fourth of the annual LIBOR on those dates plus 150 basis points: (LIBOR % + 1.5%)/4. The cap Eastern Bank is offering Gulf has the following terms:
    • Seven caplets with expiration dates of 3/20, 6/20, and 9/20.
    • The cap rate on each caplet is 2.5%.
    • The time period for each caplet is 0.25 per year.
    • The payoffs for each caplet are at the interest payment dates of the loan.
    • The reference rate is the LIBOR.
    • Notional principal is $150 million.
    • The cost of the cap is $300,000.

    T	3/20/T1	6/20/T1	9/20/T1	12/20/T1	3/20/T2	6/20/T2	9/20/T2
    X	$993,750	$993,750	$993,750	$993,750	$993,750	$993,750	$993,750
    C0	4	4.2	4.4	4.6	4.8	5	5.2
    Cost of 10 Calls:								Total Cost = $80,500
    Date	LIBOR %	Futures and Spot Price ST = fT	Call Cash Flow at Option’s Expiration 10(Max[fT − $993,750, 0]	Value of Call Cash Flow at Receipt Date (Call CF at T)(1+LIBOR).25	Quarterly Interest at Receipt Date 0.25 [(LIBOR + 0.01)] ($10m)	Hedged Interest Revenue Col 6 + Col 5	Hedged Rate [(4)(Col 7)]/$10m	Unhedged Rate LIBOR + 100bp
    12/20	1.50
    3/20	1.50	$996,250	$25,000		$62,500	$62,500.00	0.025	0.025
    6/20	2.00	$995,000	$12,500	$25,093.23	$62,500	$87,593.23	0.035	0.025
    9/20	2.50
    12/20	3.00
    3/20	3.50
    6/20	4.00
    9/20	4.50
    12/20

    Complete Table 8.3, showing the company’s quarterly interest payments, caplet cash flows, hedged interest payments (interest minus caplet cash flow), and hedged and unhedged rate as a proportion of a $150 million loan (do not include cap cost) for each period, given LIBOR rates starting at 1% on 3/1/Y1 and then increasing each period by 50 bp.

15. Southern Trust is planning to invest $150 million in a Commerce Bank two‐year floating‐rate note paying LIBOR plus 150 basis points. The floating‐rate note has a maturity of two years, starts on December 20, and is reset the next seven quarters. The initial quarterly rate is equal to 2.5%/4 and resets quarterly to equal to one fourth of the annual LIBOR on those dates plus 150 basis points: (LIBOR % + 1.5%)/4. Commerce Bank is offering Southern Trust a floor with the following terms:
    • Seven floorlets with expiration dates of 3/20, 6/20, and 9/20.
    • The floor rate on each caplet is 2.5%.
    • The time period for each caplet is 0.25 per year.
    • The payoffs for each floorlet are at the interest payment dates on the FRN.
    • The reference rate is the LIBOR.
    • Notional principal is $150 million.
    • The cost of the floor is $200,000.

    Loan: Floating Rate Loan; Term = 2 years; Reset Dates: 3/1, 6/1, 9/1, 12/1; Time Frequency = 0.25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date Cap: Cost of Cap = $300,000; Cap Rate = 2.50%; Reference Rate = LIBOR; Time Frequency = 0.25; Caplets’ Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date
    1	2	3	4	5	6	7
    Reset Date	Assumed LIBOR	Loan Interest on Payment Date (LIBOR + 150bp)(0.25)($150m)	Cap Payoff on Payment Date Max[LIBOR−0.025,0] (0.25) ($150m)	Hedged Interest Payment Col. (3) − Col. (4)	Hedged Rate 4[Col (5)/$150m]	Unhedged Rate LIBOR + 150bp
    3/1/Yn	0.010
    6/1/Y1	0.010	$937,500	$0	$937,500	0.0250	0.0250
    9/1/Y1	0.015	$937,500	$0	$937,500	0.0250	0.0250
    12/1/Y1
    3/1/Y2
    6/1/Y2
    9/1/Y2
    12/1/Y2
    3/1/Y3

    ⁿThere is no cap on this date

    Complete Table 8.4, showing the Southern Trust’s quarterly interest receipts, floorlet cash flows, hedged interest receipts (interest plus floorlet cash flow), and hedged and unhedged rate as a proportion of a $150 million investment (do not include floor cost) for each period given LIBOR rates starting at 1% on 3/1/Y1 and then increasing each period by 50 bp.

    Investment: $150 million in floating‐rate note; Term = 2 years; Reset Dates: 3/1, 6/1, 9/1, 12/1; Time Frequency = 0.25; Rate = LIBOR + 150BP; Payment Date = 90 days after reset date Floor: Cost of Floor = $200,000; Floor Rate = 2.5%; Reference Rate = LIBOR; Time Frequency = .25; Floorlets’ Expirations: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date
    1	2	3	4	5	6	7
    Reset Date	Assumed LIBOR	Interest Received on Payment Date (LIBOR + 150bp)(0.25)($150m)	Floor Payoff on Payment Date Max[0.025−LIBOR,0] (0.25)($150m)	Hedged Interest Payment Col. (3) + Col. (4)	Hedged Rate 4[Col (5)/$150m]	Unhedged Rate LIBOR + 150bp
    3/1/Y1n	0.010
    6/1/Y1	0.010	$937,500	$0	$937,500	0.025	0.025
    9/1/Y1	0.015	$937,500	$562,500	$1,500,000	0.040	0.025
    12/1/Y1
    3/1/Y2
    6/1/Y2
    9/1/Y2
    12/1/Y2
    3/1/Y3

    ⁿ There is no floor on this date

Bloomberg Exercises

1. Using the Bloomberg OSA screen, evaluate a stock insurance position and a short‐range forward contract for a selected stock. Assume you own 100 shares of the stock and plan to sell it or hedge its value at the options’ expiration. Bloomberg sequence: Stock Ticker <Equity>; OSA; load call and put options (for short‐range forward position, select a call with a higher exercise price than the put used to create your floor); on the OSA screen, select “Market Value” and the expiration date as your evaluation dates.
2. Using the Bloomberg OSA screen, evaluate a cap and long‐range forward contract on a stock purchase of a selected stock. Assume you plan to buy 100 shares of the stock at the expiration of the options. Bloomberg sequence: Stock Ticker <Equity>; OSA; load call and put options (for long‐range forward position, select a put with a lower exercise price than the call used to cap the purchase); on the OSA screen, make the stock position negative to indicate purchase (e.g., −100); on OSA screen select “Market Value” (negatives values indicate cost) and the expiration date as one of your evaluation dates.
3. Using the Bloomberg OSA screen, evaluate an insurance and short‐range forward position to hedge the futures value of 100 shares of a selected equity ETF (e.g., S&P 500 Spider, SPY). If there are listed options on the ETF’s options, use them; otherwise, use the S&P spot or futures index options and the price‐sensitivity model to determine the number of option contracts needed to set up the hedge. Bloomberg sequence: ETF Ticker <Equity>; OSA; load call and put options (for short‐range forward position, select a call with a higher exercise price than the put used to create your floor); on the OSA screen select “Market Value” and the expiration date as your evaluation dates.
4. Evaluate a future sale of a commodity such as wheat with a floor and short‐range forward positions.
   1. Select a commodity that has a futures options contracts on it.
   2. Use the SECF screen to find the tickers for futures, futures options, and spot contracts.
      • For futures options: SECF <Enter>; select “Commodities” from the “Category” dropdown; and then click the “Opt” tab.
      • For spot position: select “Commodities” from the “Category” dropdown and then select “Spot” from the “Instrument” tab.
   3. Upload the spot commodity’s menu screen: Commodity Ticker <Comdty> or Commodity Ticker <Index> (e.g., WEATTKHR <Index> for wheat).
   4. Bring up the OSA screen for the loaded commodity and select a position (e.g., 5,000 bushels to sell). Evaluate the position at different dates (Click “Scenario Chart” tab).
   5. From the red “Positions” tab on OSA, click “Add Listed Options” and then the futures ticker in the upper‐right amber area box (e.g., W A for CBT wheat futures contracts).
   6. Select the call and put options on the futures needed to evaluate the insurance and short‐range forward contracts.
   7. On the OSA screen, set the number of long puts needed to insure the commodity sale and the number of short calls needed to set up the short‐range forward position.
   8. Click “Scenario Chart” tab and input setting: profit/loss, market value, range (−20% to 20%), and evaluation dates.
5. Evaluate a future purchase of a selected commodity like crude oil with a cap and long‐range forward position using Bloomberg’s OSA screen.
   1. Select a commodity that has futures options contracts on it.
   2. Use the SECF screen to find the tickers for futures, futures options, and spot contracts.
      • For futures options: SECF <Enter>; select “Commodities” from the “Category” dropdown and then click the “Opt” tab.
      • For spot position: select “Commodities” from the “Category” dropdown and then select “Spot” from the “Instrument” tab (e.g., USCRWTIC for Crude Oil and West Texas Crude).
   3. Upload the spot commodity’s menu screen: Commodity Ticker <Comdty> or Commodity Ticker <Index> (e.g., USCRWTIC <Index> for West Texas Crude).
   4. Bring up the OSA screen for the loaded commodity and select a position (e.g., –10,000 for 10,000 barrels to buy). Evaluate the position at different dates (Click “Scenario Chart” tab).
   5. From the red “Positions” tab on OSA, click “Add Listed Options” and then the futures ticker in the upper‐right amber area box (e.g., CLA for crude oil futures).
   6. Select the call and put options on the futures needed to evaluate the cap and long‐range forward contract.
   7. On the OSA screen, set the number of calls needed to cap the cost of the commodity and the number of short puts for the long‐range forward contract.
   8. Click “Scenario Chart” tab and input setting: Market value, range (−20% to 20%), and evaluation dates. Note: Negative values indicate the cost of the commodity.
6. Access Bloomberg information on a currency futures contract with options: type CTM to bring up the “Contract Table Menu,” click “Categories” and “Currency,” search for CME‐listed futures on the Menu (type CME in the amber Exchange box area), find the CME contract of interest and bring up the contract’s menu screen (Ticker <Comdty>; e.g., BPA <Curncy> for British pound futures), and then use the Expiration screen (EXS) on the contract’s menu page to find the ticker for the contract with the expiration month you want to analyze (Ticker <Curncy>; e.g., BPH7 <Curncy> for the March 2017 contract). View screens to examine: DES, DLV, and GP.
7. On the futures’ screen, use OSA to load the currency futures’ options contracts. Using OSA, generate profit graphs for the following positions:
   1. Long position in the futures and futures put
   2. Short position in the futures and futures call
   3. Long position in the futures and short position in the call
   4. Short position in the futures and short position in the put
8. Evaluate an equity mutual fund position with at least 10,000 shares that is to be sold or valued at a future date and hedged with a floor and short‐range forward positions.
   1. Use the SECF screen to find the fund: SECF <Enter>; select “Funds” from the “Category” dropdown and then click the “Open End” tab and select Equity from the “Focus” dropdown. You may want to limit your search to US funds that are dollar‐denominated and with a broad‐based or narrow‐based fund objective.
   2. Upload the fund’s menu screen and evaluate its features (include its beta) using PORT (“Characteristics” and “Summary” tabs) or HRA.
   3. Bring up the OSA screen for the loaded fund and select a position (e.g., 10,000 shares). Evaluate the position at different dates (Click “Scenario Chart” tab).
   4. From the red “Positions” tab on OSA, click “Add Listed Options” and then type in the ticker for the S&P 500 (SPX) or S&P 500 futures (SPA), or another selected equity spot or futures index with options, in the upper‐right amber area box.
   5. Select the call and put options on the futures needed to evaluate the insurance and short‐range forward contracts. Use the price‐sensitivity model to determine the number of options.
   6. On the OSA screen, set the number of long puts needed to ensure the portfolio and the number of short calls for the short‐range forward contract.
   7. Click “Scenario Chart” tab and input setting: profit/loss, market value, range (−20% to 20%), and evaluation dates.

   Using the OSA screen, evaluate a call‐enhanced strategy. On the OSA screen, input a number of long call contracts to enhance the fund’s value if the market increases.

9. Construct your own equity portfolio and then analyze it using PORT and also as an index created using the CIXB screen. Guidelines:
   • In constructing your own portfolio, use the Equity Search screen (EQS), the FMAP screen to identify stocks that make up different types of funds, or the SECF screen to identify stocks making up a fund or equity index.
   • Make the number of shares for the stocks in your portfolio large enough so that your portfolio’s market value is at least $10 million.
   • Create historical data for your portfolio. See sections in Chapter 3: “Bloomberg: Equity Index Futures and Related Screens” and “Steps for Creating Data in PRTU.”
   • Import your portfolio in CIXB and create historical data. See sections in Chapter 3: “Bloomberg: Equity Index Futures and Related Screens” and “Bloomberg CIXB and OSA Screens.”
10. Using the Bloomberg OSA screen, evaluate an insurance and short‐range forward position to hedge the futures value of the portfolio you constructed in Exercise 9 (or another portfolio you have constructed with a market value of at least $10 million). Use S&P 500 spot or futures options (for the short‐range forward position, select a call with a higher exercise price than the put’s exercise price used to create your floor), and use the price‐sensitivity model to determine the number of options. See the section “Bloomberg Hedging Screens” for a guide in loading portfolios and options in OSA.

    Using the OSA screen, evaluate a call‐enhanced strategy. On the OSA screen, input a number of long call contracts to enhanced the portfolio’s value if the market increases.

11. Access Bloomberg information on a T‐notes futures contract with options. Bloomberg sequence: type CTM to bring up the “Contract Table Menu,” click “Categories” and “Bonds,” search for CBT‐listed contracts on the menu (type CBT in the amber Exchange box area), find the CBT contract of interest, and bring up the contract’s menu screen (Ticker <Comdty>; e.g., FVA <Comdty> for five‐year Treasury), and then use the expiration screen (EXS) on the contract’s menu page to find the ticker for the contract with the expiration month you want to analyze (Ticker <Comdty>; e.g., FVH7 <Comdty> for the March 2017 contract). View screens to examine: DES, DLV, and GP.
12. Using the OSA screen, generate profit graphs on the futures options you selected in Exercise 11 for the following positions:
    1. Long position in the futures and futures put
    2. Short position in the futures and futures call
    3. Long position in the futures and short position in the call
    4. Short position in the futures and short position in the put
13. Using the Bloomberg MARS screen, load the futures, futures put, futures call, and cheapest‐to‐deliver notes on the T‐note futures contract you analyzed in Exercise 11. For a guide, see sections “Bloomberg Hedging Screens” and “MARS: Bond Hedging with Futures and Futures Options Using the MARS Platform.” Using MARS, evaluate and compare the following positions on the futures options’ expiration date for interest rate shifts ranging between –50 bps and +50 bps:
    1. Unhedged bond position
    2. Bond position hedged with a short futures contract
    3. Bond position hedged with a long futures put contract
    4. Bond purchase position evaluated at the expiration date on the futures or option (make the bond position negative to reflect cost)
    5. Bond purchase hedged with a long futures contract
    6. Bond purchase hedged with a long futures call contract
14. Construct a portfolio of investment‐grade corporate bonds and US Treasuries using the PRTU screen. After constructing the bond fund, evaluate the portfolio in PORT. Guidelines:
    • Use the Bloomberg search/screen function, SRCH, or SECF to identify the bonds for your portfolio.
    • Make the number of issues for the bonds in your portfolio large enough so that your portfolio’s market value is at least $10 million.
    1. Evaluate the characteristic of your portfolio in PORT: “Characteristics” tab, “Summary View.”
    2. Access Bloomberg information on a T‐notes futures contract with options that has a maturity close to the average maturity of your portfolio. Bloomberg sequence: type CTM to bring up the “Contract Table Menu,” click “Categories” and “Bonds,” search for CBT on the Menu (type CBT in the amber Exchange box area), find the CBT contract of interest, and bring up the contract’s menu screen (Ticker <Comdty>; for example, FVA <Comdty> for five‐year Treasury), and then use the expiration screen (EXS) on the contract’s menu page to find the ticker for the contract with the expiration month you want to analyze (Ticker <Comdty>; e.g., FVH7 <Comdty> for the March 2017 contract). View screens to examine: DES, YAS, DLV, and GP.
15. Using the Bloomberg MARS screen, load the futures, futures put, futures call, and bond portfolio you analyzed in Exercise 14. For a guide, see sections “Bloomberg Hedging Screens” and “MARS: Importing Bond Portfolios into Bloomberg’s MARS Screen and Adding Options—Bond Portfolio Insurance and Changing Market Exposure.” Use the price‐sensitivity model to determine the number of futures options contracts you would need to hedge the value of the portfolio. Using MARS, evaluate and compare the following positions on the futures options’ expiration date for interest rate shifts ranging between –50 bps and +50 bps:
    1. Unhedged bond portfolio position
    2. Bond portfolio position hedged with short futures contracts
    3. Bond portfolio position hedged with long futures put contracts
    4. Bond portfolio position with an enhanced exposure to interest rate changes with long futures positions
    5. Bond portfolio position with an enhanced exposure to interest rate changes with long futures call positions
16. Access Bloomberg information on a CME Eurodollar futures contract. Bloomberg sequence: Type CTM to bring up the “Contract Table Menu,” click “Categories” and “Interest Rates,” search for CME‐listed contracts on the menu screen (type CME in the amber Exchange box), find the CME contract of interest, and bring up the contract’s menu screen (Ticker <Comdty>; e.g., EDA <Comdty> for the 3‐month Eurodollar contract), and then use the expiration screen (EXS) on the contract’s menu page to find the ticker for the contract with the expiration month you want to analyze (Ticker <Comdty>; e.g., EDH7 <Comdty> for the March 2017 contract). View screens to examine DES and GP.
17. On the futures screen, use OSA to load the futures’ options contracts for the options you selected in Exercise 16. Using OSA, generate profit graphs for the following positions:
    1. Long position in the futures and futures put
    2. Short position in the futures and futures call
    3. Long position in the futures and short position in the call
    4. Short position in the futures and short position in the put